Clustering and Classification
date()## [1] "Thu Nov 24 22:09:25 2022"Read the data
library(MASS)
# load the data
data("Boston")
# explore the dataset
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14View(Boston)This data consists of housing values in Suburbs of Boston. It has 506 observations and 14 variables. The variables are as follows : crim (per capita crime rate by town), zn (proportion of residential land zoned for lots over 25,000 sq.ft.), indus (proportion of non-retail business acres per town.), chas (Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)), nox (nitrogen oxides concentration (parts per 10 million)), rm (average number of rooms per dwelling.), age (proportion of owner-occupied units built prior to 1940), dis (weighted mean of distances to five Boston employment centres), rad (index of accessibility to radial highways), tax (full-value property-tax rate per $10,000), ptratio (pupil-teacher ratio by town), black (1000(Bk - 0.63)^21000(Bk−0.63) where BkBk is the proportion of blacks by town, lstat (lower status of the population (percent)), medv(median value of owner-occupied homes in $1000s).
None of the variables are categorical. All of them are numerical or integers.
Graphical overview and summaries of the variables
#Summary of the variable
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00The Chas is a binary variable and the variable rad is an index variable. Some variables have a higher variability compared to some others. For example, the variable tax (full-value property-tax) ranges from 187 to 711 and the variable black (proportion of blacks by town) ranges from 0.32 to 396.90. However, the variable nox (nitrogen oxides concentration) ranges only from 0.3850 to 0.8710 and the variable rm (average number of rooms per dwelling) ranges only from 3.561 to 8.780.
# Graphical exploration of data
pairs(Boston)
Even though, this graph is somewhat complicated at first, we can get a rough idea about how the variables are. The correlation plot can be used for further understanding about the data.
library(tidyr)
library(corrplot)## corrplot 0.92 loaded# correlation matrix
cor_matrix <- cor(Boston)
cor_matrix %>% round(digits = 2)## crim zn indus chas nox rm age dis rad tax ptratio
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58 0.29
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31 -0.39
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72 0.38
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04 -0.12
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67 0.19
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29 -0.36
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51 0.26
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53 -0.23
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91 0.46
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00 0.46
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46 1.00
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44 -0.18
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54 0.37
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47 -0.51
## black lstat medv
## crim -0.39 0.46 -0.39
## zn 0.18 -0.41 0.36
## indus -0.36 0.60 -0.48
## chas 0.05 -0.05 0.18
## nox -0.38 0.59 -0.43
## rm 0.13 -0.61 0.70
## age -0.27 0.60 -0.38
## dis 0.29 -0.50 0.25
## rad -0.44 0.49 -0.38
## tax -0.44 0.54 -0.47
## ptratio -0.18 0.37 -0.51
## black 1.00 -0.37 0.33
## lstat -0.37 1.00 -0.74
## medv 0.33 -0.74 1.00# visualize the correlation matrix
corrplot(cor_matrix, method="square", type="upper", cl.pos = "b", tl.pos = "d", col = COL2('PiYG'), addCoef.col = 'black', tl.cex = 0.6)
The bigger and more colourful the square in the cell is, the stronger the correlation is between the variables. The purple colour of the square indicates negative correlation while the green colour indicates a positive correlation. The highest positive correlation is between the variables: tax (full-value property-tax rate per $10,000) and rad (index of accessibility to radial highways).There is a 0.91 correlation between those two variables. The strongest negative correlation is between the varioables: nox (nitrogen oxides concentration (parts per 10 million)) and dis (weighted mean of distances to five Boston employment centres). There is a -0.77 correlation between those two variables. There is a 0.91 correlation between those two variables.
Standardize the dataset
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- as.data.frame(scale(Boston))Each and every mean of the summary of the scaled dataset is zero. It shows that after the standardization, all variables fit to a normal distribution.
# Create a categorical variable of the crime rate
boston_scaled$crim <- as.numeric(boston_scaled$crim)
bins <- quantile(boston_scaled$crim)
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE,labels = c("low","med_low","med_high","high"))
table(crime)## crime
## low med_low med_high high
## 127 126 126 127# Drop the crim variable and add crime variable
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)
# Divide the dataset
n <- nrow(boston_scaled)
ind <- sample(n, size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]Fit the linear discriminant analysis
# Creating the model
lda.fit <- lda(crime ~ ., data = train)
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2475248 0.2475248 0.2524752 0.2524752
##
## Group means:
## zn indus chas nox rm age
## low 0.9439986 -0.8561482 -0.07547406 -0.8713159 0.46156803 -0.8442935
## med_low -0.1489396 -0.2074352 -0.03610305 -0.5170630 -0.15257867 -0.2321906
## med_high -0.3930786 0.1881207 0.22945822 0.4397051 0.03174814 0.4261169
## high -0.4872402 1.0171096 -0.04073494 1.0630805 -0.35195732 0.8178021
## dis rad tax ptratio black lstat
## low 0.8225070 -0.6855667 -0.7644438 -0.42381371 0.38136174 -0.75853754
## med_low 0.2510793 -0.5511961 -0.4468876 -0.03073223 0.31423748 -0.07019450
## med_high -0.3637260 -0.4053858 -0.2896429 -0.24845269 0.04814488 0.03604884
## high -0.8439063 1.6382099 1.5141140 0.78087177 -0.86588545 0.95020413
## medv
## low 0.54867202
## med_low -0.03216299
## med_high 0.09471625
## high -0.73589473
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.067991911 0.788282707 -0.87403685
## indus 0.007873984 -0.080004355 0.24316158
## chas -0.068535208 -0.046514046 0.05729580
## nox 0.390982609 -0.774084361 -1.43956310
## rm -0.090522299 0.004810719 -0.07950965
## age 0.250902992 -0.365549046 -0.13100452
## dis -0.028386691 -0.365514208 0.09467384
## rad 3.225231420 1.016277534 -0.24607817
## tax -0.047738699 -0.216902384 0.77129621
## ptratio 0.147369406 -0.006510985 -0.32895033
## black -0.130430927 0.031804843 0.11693427
## lstat 0.222480353 -0.140012109 0.45392594
## medv 0.185418675 -0.381530212 -0.25431491
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9500 0.0364 0.0136# Visualization of the model
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
classes <- as.numeric(train$crime)
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 2)
The Linear Discriminant Analysis was able to separate the high crime classes well from the other classes (low, med_low, med_high). The most influential line separator is the rad: index of accessibility to radial highways. On the other hand, there is a clear separation between low crime class and med_high crime class which were caused by zn (proportion of residential land zoned for lots over 25,000 sq.ft.) and the nox (nitrogen oxides concentration (parts per 10 million)). This may be due to the differences in rural and urban setting.
The first discriminant function separates 95.25% of the population, while the second discriminant function separates 3.75% of the population. The third discriminant function separates only 1% of the population.
Prediction with the LDA Model
# Save the correct classes and remove the criome variables from the test data
correct_classes <- test$crime
test <- dplyr::select(test, -crime)#predict with the created model
lda.pred <- predict(lda.fit, newdata = test)
#perform cross tabulation
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 19 8 0 0
## med_low 9 15 2 0
## med_high 0 7 16 1
## high 0 0 0 25The model predicts the high crime class well. The model didn’t predict the low crime class well. Out of 102 observations, 71% of them were correctly predicted. Thus, the model can be used for further prediction purposes.
Clustering
# Reload and scale data set
data(Boston)
boston_scaled <- scale(Boston)
# Create euclidean distance matrix
dist_eu <- dist(boston_scaled)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# Create Manhattan distance matrix
dist_man <- dist(boston_scaled, method = "manhattan")
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618The distances of each method gives us a different result.
# k-means clustering
km <- kmeans(boston_scaled, centers = 3)
# plot the Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)
# Optimal number of clusters
set.seed(123)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
# Visualization
library(ggplot2)
qplot(x = 1:k_max, y = twcss, geom = 'line')## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
According to the above graph, two clusters would be the optimal number of clusters in this case.
# k-means clustering for 2 clusters
km <-kmeans(boston_scaled, centers = 2)
# plot the normalized Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)
km <- kmeans(boston_scaled, centers = 2)
pairs(boston_scaled[,c(1,2,3,4,5,6,7)], col = km$cluster)
km <- kmeans(boston_scaled, centers = 2)
pairs(boston_scaled[,c(8,9,10,11,12,13,14)], col = km$cluster)
The pairs plot shows a clear separation of two populations in some variables. One cluster is associated with low crimes, low proportion of non-retail business acres per town, low nitrogen oxides concentration, lower age, and high median value of owner-occupied homes.
Bonus
# k-means clustering
km2 <- kmeans(boston_scaled, centers = 3)
# plot the Boston dataset with clusters
pairs(boston_scaled, col = km2$cluster)
# linear discriminant analysis
boston_scaled <- data.frame(scale(Boston))
lda.fit2 <- lda(km2$cluster ~ ., data = boston_scaled)
# print the lda.fit object
lda.fit2## Call:
## lda(km2$cluster ~ ., data = boston_scaled)
##
## Prior probabilities of groups:
## 1 2 3
## 0.4664032 0.3241107 0.2094862
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.3760908 -0.3417123 -0.296848 0.01127561 -0.3345884 -0.09228038
## 2 0.8046456 -0.4872402 1.117990 0.01575144 1.1253988 -0.46443119
## 3 -0.4075892 1.5146367 -1.068814 -0.04947434 -0.9962503 0.92400834
## age dis rad tax ptratio black
## 1 -0.02966623 0.05695857 -0.5803944 -0.6030198 -0.08691245 0.2863040
## 2 0.79737580 -0.85425848 1.2219249 1.2954050 0.60580719 -0.6407268
## 3 -1.16762641 1.19486951 -0.5983266 -0.6616391 -0.74378342 0.3538816
## lstat medv
## 1 -0.1801190 0.03577844
## 2 0.8719904 -0.68418954
## 3 -0.9480974 0.97889973
##
## Coefficients of linear discriminants:
## LD1 LD2
## crim -0.03134296 0.14880455
## zn -0.06381527 1.22350515
## indus 0.61086696 0.10402980
## chas 0.01953161 -0.03579238
## nox 1.00230143 0.70464917
## rm -0.16285767 0.44390394
## age -0.07220634 -0.59785382
## dis -0.04270475 0.45498614
## rad 0.71987743 0.02882054
## tax 0.98285440 0.70663319
## ptratio 0.22527977 0.15514668
## black -0.01693595 -0.03181845
## lstat 0.18274033 0.50122677
## medv -0.02892966 0.64244841
##
## Proportion of trace:
## LD1 LD2
## 0.8409 0.1591# Visualization of the model
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
classes <- as.numeric(km2$cluster)
plot(lda.fit2, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit2, myscale = 3)
The clusters have separated very well. The nox (nitrogen oxides concentration), zn (proportion of residential land zoned for lots over 25,000 sq.ft.) and age (proportion of owner-occupied units built prior to 1940= seem to be the most influential line seperators.
Super Bonus
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)# Visualization of the 3D Plot
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')# Visualization of the 3D Plot (crime classes as colours)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)# Visualization of the 3D Plot (k mean clusters as colours)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km$cluster[ind])The first plot shows a well separated plot which has only two visible clusters. In the second plot, high crime class has a separate cluster by their own. All the other classes seem to mix with each other. Again in the third plot, there are two well separated clusters.